How Random Connectivity Shapes the Fluctuating Dynamics of Finite-Size Neural Populations

Mesoscopic models of finite-size neuronal populations are crucial to understand the dynamics of neural networks in the brain, especially their fluctuations and response to stimuli. However, current theories to derive such models are based on the assumption of homogeneous all-to-all (full) connectivity, neglecting the variance in the connectivity of biologically realistic networks with connection probabilities <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>p</a:mi> <a:mo>&lt;</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> (nonfull connectivity). To gain insight into the different fluctuation mechanisms underlying the neural variability of populations of spiking neurons, we derive and analyze a stochastic mean-field model for finite-size networks of Poisson neurons with random connectivity (including nonfull connectivity), external noise, and disordered mean inputs. We treat the quenched disorder of the connectivity by an annealed approximation enabling a doubly stochastic description of synaptic inputs for finite network size. A further reduction leads to a low-dimensional closed system of coupled Langevin equations for the mean and variance of the neuronal membrane potentials as well as a variable capturing finite-size fluctuations arising specifically in the case of connectivity disorder. Comparing to microscopic simulations, we find that the mesoscopic model describes the fluctuations and nonlinearities well and outperforms previous mesoscopic models that neglected the variance in the connectivity. The joint effect of connectivity disorder and finite network size can be analytically understood by a softening of the effective nonlinearity and the multiplicative character of spiking noise. The mesoscopic theory shows that quenched disorder can stabilize the asynchronous state, and it correctly predicts large quantitative and nontrivial qualitative effects of connection probability on the variance of the population firing rate and its dependence on stimulus strength. Our theory thus elucidates how disordered connectivity shapes nonlinear dynamics and fluctuations of neural populations at the mesoscopic scale and showcases a useful mean-field method to treat random connectivity in finite-size, spiking neural networks.